What is the difference between extrapolation and interpolation, and what is the most correct way to use these terms?

For example, I saw a statement in an article that used interpolation like:

“Process to interpolate the shape of the estimator between bin points”

A sentence that uses both extrapolation and interpolation is, for example:

In the previous step, we extrapolate the interpolation function using the Kernel method in the left and right temperature tails.

Watching: What is Extrapolation?

Can someone provide a clear and easy way to differentiate them and guide how to use these terms correctly with an example?

terminology interpolation extrapolation

— Frank Swanton source


A related question.

— JM is not a statistician


Possible duplicate of What is wrong with extrapolation?

— usεr11852 says Monic Recovery

usεr11852 I think the two questions cover similar but different ground because this question asks for contrast with interpolation.

— mkt – Recovery

This distinction between interpolation and extrapolation has been strictly formalized in a generally agreed manner, (e.g., through convex hulls) or are these terms still subject to judgment? and human interpretation?

— Nick Alger



To add a visual explanation for this: consider a few points where you plan to model.

They look like they can be well described with a straight line, so you fit a linear regression to them:

This regression line allows you to both interpolate (generate expected values ​​in the middle of your data points) and extrapolate (generate expected values ​​outside the range of your data points). friend). I highlighted the extrapolation in red and the largest interpolation area in blue. For clarity, even small regions between points are interpolated, but I’ve only highlighted large regions here.

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Why is extrapolation in general a more concern? Because you are often less certain about the shape of the relationship outside of your data range. Consider what can happen when you collect a few more data points (empty circles):

It turns out that the relationship wasn’t captured well with your hypothetical relationship after all. The predictions in the extrapolation area are off. Even if you correctly guessed the function that correctly describes this non-linear relationship, your data doesn’t scale enough for you to get a good grasp of the nonlinearity, so you might still be pretty far away. Note that this is a problem not only for linear regression, but for any relationship – this is why extrapolation is considered dangerous.

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Predictions in the interpolation area are also inaccurate because of the lack of nonlinearity in the fit, but their prediction error is much lower. There’s no guarantee that you won’t have an unexpected relationship in between your points (i.e. interpolated area), but generally less likely.

I would add that extrapolation is not always a bad idea – if you extrapolate a little bit outside the range of your data, you probably won’t be very wrong (although it is possible). !). Ancient people without a good scientific model of the world would not be wrong if they predicted that the sun would rise again the next day and the day after (though one day in the future, even this will failure).

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Editing based on comments: whether interpolating or extrapolating, it’s always best to have some theory to grounding expectations. If theoretical modeling must be performed, the risk from interpolation is usually less than from extrapolation. That said, as the distance between data points increases in magnitude, interpolation also becomes more and more dangerous.